关于Taylor中值函数若干分析性质的讨论

给出了"Taylor中值函数"的定义,对Taylor中值函数的分析性质进行了系统的综合讨论,证明了Taylor中值函数的单调性、可积性、连续性、可微性等分析性质.     

积分型Cauchy中值函数若干分析性质

给出"积分型Cauchy中值函数"的定义,对"积分型Cauchy中值函数"的分析性质进行了系统讨论,证明了"积分型Cauchy中值函数"的单调性、可积性、连续性、可微性等分析性质.作为"积分型Cauchy中值函数"的特例,给出了"第一积分中值函数"的定义及"第一积分中值函数"相应的分析性质.     

几类中值定理中间点的分析性质

给出了广义Taylor公式、高阶Cauchy中值定理及加权型中值定理中间点的单值性、连续性及可导性的充分条件,并给出了求导公式.     

第二积分中值函数

通过上、下确界定义,给出了"第二积分中值函数"的定义,并对"第二积分中值函数"的单调性、可积性、连续性、可导性等分析性质进行了系统的讨论.     

第一积分中值函数

通过上下确界,给出了"第一积分中值函数"的定义,对"第一积分中值函数"的分析性质进行了系统的讨论,证明了"第一积分中值函数"的单调性、可积性、连续性、可导性等分析性质.     

关于广义Taylor中值定理中ξ的渐近性

本讨论了当区间两端点都趋向于一定点时,广义Taylor中值定理中ξ的渐近性。     

二元函数微分中值定理中值点的分析性质

讨论二元函数微分中值定理中值点的连续性及可导性问题,给出二元函数微分中值定理中值点连续及偏导数存在的充分务停,同时给出计算其偏导数的公式。     

积分第一中值定理的推广

将积分第一中值定理中的连续性条件减弱为有介值性,建立了具有介值性质的可积函数的积分第一中值定理的推广形式.     

积分中值定理的渐进性推广

基于积分中值定理和推广的积分中值定理。通过构造辅助函数．借助罗必达法则。可以得出当区间长度趋于0时推广的积分第一中值定理中值点的渐近性描述．渐近性质的可导性条件可减弱为极限存在性条件，其参数要求也可由非零自然数推广到实数．     

广义拟可微函数的微分

本文给出一种广义拟可微函数类,它是Demyanov与Rlubinov(1980)意义下拟可微函数的推广,通过凸集类对的空间的某些理论,建立了这类广义拟可微函数的微分学理论,包括加法运算、数乘运算、乘法运算、除法运算、极大值运算,极小值运算以及复合运算的微分公式和中值定理。这些结果为广义拟可微类函数优化研究提供了基本工具．     

广义泰勒定理:"同伦分析方法"之有效性的一个数理逻辑证明

推导了复变函数一个广义意义上的泰勒级数表达式,证明了有关的收敛性定理,大大增大摄动级数解的收敛区域。定理的证明亦为一种新的、求解非线性问题的解析方法（即“同伦分析方法”）的有效性奠定了一个坚实的数理逻辑基础。     

Power series with integer coefficients in several variables

A classical theorem of Borel-Pólya, which concerns rationality of an analytic function whose Taylor expansion at a point has integer coefficients, is generalized to several variables. This research partially supported by an Indiana University Summer Faculty Fellowship.     

The essence of the generalized Taylor theorem as the foundation of the homotopy analysis method

The generalized Taylor theorem is the foundation of the homotopy analysis method proposed by Liao. This theorem is interesting but hard to understand from the mathematical point of view. Especially, there is a key parameter h whose meaning is still unknown. In the paper, we derive the generalized Taylor theorem from a usual way, that is, we prove that the generalized Taylor expansion is equivalent to a different representation of the usual Taylor expansion at different points. Therefore the meaning of the auxiliary parameter h is clarified. These results give a reasonable explanation of the parameter h and uncover the essence of the generalized Taylor theorem from which we can deeply understand the homotopy analysis method. Through the detailed analysis of some examples, we compare the series solution at the different points with the generalized Taylor series solution obtained by the homotopy analysis method.     

Basics on growth orders of polymonogenic functions

In this paper, we introduce generalizations of the classical growth order and the growth type of analytic functions in the context of polymonogenic functions. Polymonogenic functions are null-solutions of higher integer order iterates of a generalized higher dimensional Cauchy–Riemann operator. One of the main goals is to prove generalizations of the famous Lindelöf–Pringsheim theorem linking explicitly these growth orders and growth types with the Taylor series coefficients in the context of this function class.     

Best proximity points: global minimization of multivalued non-self mappings

In this article, we give a best proximity point theorem for generalized contractions in metric spaces with appropriate geometric property. We also, give an example which ensures that our result cannot be obtained from a similar result due to Amini-Harandi (Best proximity points for proximal generalized contractions in metric spaces. Optim Lett, 2012). Moreover, we prove a best proximity point theorem for multivalued non-self mappings which generalizes the Mizoguchi and Takahashi’s fixed point theorem for multivalued mappings.     

Uniqueness theorem for linear elliptic equation of the second order with constant coefficients

The interior uniqueness theorem for analytic functions was generalized by M. B. Balk to the case of polyanalytic functions of order n. He proved that if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. In this paper the interior uniqueness theorem is generalized to the solution to a linear homogeneous second order differential equation of elliptic type with constant coefficients.     

Pseudocontinuability and properties of analytic functions on boundary sets of positive measure

The following analogue of Fabry's theorem is proved. Assume that a function , analytic in the polydisc, has a sufficiently lacunary Taylor series. If on a subset of the torus, of positive Lebesgue measure, the function coincides in a certain sense with a function analytic in a sufficiently large subset of, then is analytic in the polydisc for some r >1. As a consequence one obtains that a nonconstant function, analytic in a ball and having a sufficiently lacunary Taylor series, cannot have angular boundary values equal in modulus to unity or having zero real part on a set of positive measure.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 88–96, 1983.     

Weyl type theorems for operators satisfying the single-valued extension property

Let T be a bounded linear operator acting on a Banach space X such that T or its adjoint T^∗ has the single-valued extension property. We prove that the spectral mapping theorem holds for the B-Weyl spectrum, and we show that generalized Browder's theorem holds for f(T) for every analytic function f defined on an open neighborhood U of σ(T). Moreover, we give necessary and sufficient conditions for such T to satisfy generalized Weyl's theorem. Some applications are also given.     

On the coefficients of a function analytic in the unit disc having slow rate of growth

Summary For a function f, analytic in z < 1, the notions of generalized orders are introduced that are specially suited for the study of the growth of f, if it is of slow growth. The characterizations of the generalized orders are found in terms of the coefficients in the Taylor series development of f. A decomposition theorem is proved for functions of hirregular growth. The results of the present paper generalize the results in [G. P. Kapoor - K. Gopal, J. Math. Anal. Appl., 74 (1980), pp. 446–455].     

Some Remarks on the Algebra of Bounded Dirichlet Series

We consider here the algebra of functions which are analytic and bounded in the right half-plane and can moreover be expanded as an ordinary Dirichlet series. We first give a new proof of a theorem of Bohr saying that this expansion converges uniformly in each smaller half-plane; then, as a consequence of the alternative definition of this algebra as an algebra of functions analytic in the infinite-dimensional polydisk, we first observe that it does not verify the corona theorem of Carleson; and then, we give in a deterministic way a new quantitative proof of the Bohnenblust-Hille optimality theorem, through the construction of a generalized Rudin-Shapiro sequence of polynomials. Finally, we compare this proof with probabilistic ones.